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Quantum computing and computational law

Jeffery Atik & Valentin Jeutner

To cite this article: Jeffery Atik & Valentin Jeutner (2021): Quantum computing and computational

law, Law, Innovation and Technology, DOI: 10.1080/17579961.2021.1977216

To link to this article: https://doi.org/10.1080/17579961.2021.1977216

© 2021 The Author(s). Published by Informa

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Quantum computing and computational law

Jeﬀery Atik

a

and Valentin Jeutner

b

a

Loyola Law School, Loyola Marymount University, Los Angeles, CA, USA;

b

Faculty of Law,

Lund University, Lund, Sweden

ABSTRACT

Quantum computing technology will greatly enhance the abilities of the

emerging ﬁeld of computational law to express, model, and operationalise

law in algorithmic form. Foreshadowing the harnessing of the power of

quantum computing technology by the legal sector, this essay targets, with

reference to computational complexity theory, the categories of

computational problems which quantum computers are better equipped to

deal with than are classical computers (‘quantum supremacy’).

Subsequently, the essay demarcates the possible contours of legal

‘quantum supremacy’by showcasing three anticipated legal ﬁelds of

quantum technology: optimisation problems, burdens of proof, and

machine learning. Acknowledging that the exact manifestation of quantum

computing technology in the legal sector is as yet diﬃcult to predict, the

essay posits that the meaningful utilisation of quantum computing

technology at a later stage presupposes a creative imagination of possible

use-cases at the present.

ARTICLE HISTORY Received 19 October 2020; Accepted 21 March 2021

KEYWORDS Quantum law; quantum computer; quantum mechanics; computational law; innovation

policy

1. Introduction

In 1998, John Preskill asked two fundamental questions concerning

quantum computers: do we want to build them, and can we build

them?

1

Twenty years later, as the ﬁrst quantum computers are being con-

structed, these questions have been answered in the aﬃrmative and the

identiﬁcation of use cases for quantum computing is in full swing.

2

Over

© 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group

This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDer-

ivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distri-

bution, and reproduction in any medium, provided the original work is properly cited, and is not altered,

transformed, or built upon in any way.

CONTACT Jeﬀery Atik jeﬀery.atik@lls.edu Burns 343, 919 Albany St., Los Angeles, CA

90015, USA; Valentin Jeutner valentin.jeutner@jur.lu.se Faculty of Law, University of Lund, 22100

Lund, Sweden

1

John Preskill, ‘Quantum Computing: Pro and Con’(1998) 454 Proceedings of the Royal Society of London

A469, 469.

LAW, INNOVATION AND TECHNOLOGY

https://doi.org/10.1080/17579961.2021.1977216

the next few years, both the European Union

3

and the United States

4

have

committed to set aside 1 billion Euros and 1.2 billion USD, respectively, to

support quantum research. Meanwhile, car manufacturers are teaming up

with technology companies to enhance the eﬃcacy of autonomous

systems (e.g. VW and Google,

5

Daimler and IBM Q

6

) while pharma-

ceutical ﬁrms aim to utilise quantum computers to dramatically enhance

the abilities of AI-supported pattern-recognition in the interest of more

precise patient diagnostics.

7

These initiatives are not only driven by the

expectation that quantum computers are able to perform tasks currently

performed by classical computers more eﬃciently (‘quantum advantage’),

but also anticipate that quantum computers will be able to perform

tasks that classical computers cannot perform at all (‘quantum

supremacy’).

8

Against this background, this essay asks: what, if anything, will quantum

computing bring to law?

9

Speciﬁcally, this essay considers the potential of

quantum computing to facilitate the emergence of computational law –the

expression, application and analysis of law in algorithmic form.

10

The challenge today, in peering forward into the age of quantum compu-

ters, is to imagine legal applications that can actually harness ‘quantum

supremacy’since there is little or no reason (beyond exercising technological

competencies) to task quantum computers with solving problems that clas-

sical computers can already successfully deal with. Thus, it is one of the aims

2

Francesco Bova, Avi Goldfarb and Roger G Melko, ‘Commercial Applications of Quantum Computing’

(2021) 8 EPJ Quantum Technology 1.

3

High Performance Computing and Quantum Technology Unit, ‘Quantum Technologies Flagship’(Digital

Single Market - European Commission, 12 May 2016) <https://ec.europa.eu/digital-single-market/en/

quantum-technologies> accessed 21 June 2021.

4

Martin Giles, President Trump has signed a $1.2 billon law to boost US quantum tech,MIT Technology

Review,https://www.technologyreview.com/f/612679/president-trump-has-signed-a-12-billon-law-to-

boost-us-quantum-tech/ (last visited Aug 4, 2019).

5

Volkswagen, ‘Volkswagen Group and Google Work Together on Quantum Computers’(7 November

2017) <media.vw.com> accessed 21 June 2021.

6

Svenja Gelowicz, ‘IBM und Daimler entwickeln Quantencomputer’(14 November 2017) <https://www.

automobil-industrie.vogel.de/ibm-und-daimler-entwickeln-quantencomputer-a-671718/> accessed 21

June 2021.

7

Esther O’Sullivan, ‘Quantum Computing, Artiﬁcial Intelligence and Health Care’(BMJ Technology Blog,3

November 2017) <https://blogs.bmj.com/> accessed 21 June 2021.

8

According to John Preskill ‘quantum supremacy’exists when ‘controlled quantum systems’can perform

tasks that ‘ordinary digital computers’cannot. Sometimes the terms ‘quantum supremacy’and

‘quantum advantage’are used interchangeably. John Preskill, ‘Quantum Computing and the Entangle-

ment Frontier’[2011] Proceedings of the 25th Solvay Conference on Physics 1, 2.

9

There is a rich literature involving quantum theory to stimulate thinking about law. For examples, see

Ted Sichelman, ‘Quantifying Legal Entropy’(2021) 9 Frontiers in Physics; William HJ Hubbard, ‘Quantum

Economics, Newtonian Economis, and Law’[2017] Michigan State Law Review 425; Henry E Smith,

‘Modularity in Contracts: Boilerplate and Information Flow’(2006) 104 Michigan Law Review 1175.

10

For a comprehensive overview of computational law, see Kevin D. Ashley, Artiﬁcial Intelligence and

Legal Analytics - New Tools for Law Practice in the Digital Age (Cambridge University Press, 2017); Natha-

niel Love and Michael Genesereth, ‘Computational Law’[2005] Proceedings of the 10th International

Conference on Artiﬁcial Intelligence and Law 205.

2J. ATIK AND V. JEUTNER

of this essay to address this challenge by explaining which kind of problems

might be able to harness ‘quantum supremacy’.

When attempting to identify instances of ‘quantum supremacy’, compu-

tational complexity theory is a helpful point of departure. Computational

complexity theory describes a given problem in terms of the classical com-

puting resources required to solve it.

11

All problems (other than those

with constant solutions) increase in complexity as the number of factors

increases. Problems can be classiﬁed into a hierarchy of complexity cat-

egories. Simpler problems are linear in nature: the amount of demanded

resources increases in a linear relationship to the number of factors. These

are easily managed by classical computers. If we ask a computer to output

the number of boots required by an army, the problem does not become

meaningfully more diﬃcult as the number of soldiers increases from

100,000–10,000,000; the number of needed boots increases in a linear pro-

portion to the number of soldiers (that is, two times the number of soldiers).

The computer may require a bit more time to calculate the required number

of boots as the number of soldiers increases, but the required computational

energy should not increase by more than a ﬁxed proportion.

More diﬃcult problems require polynomial resources. This category is

known as P, where there is a known, eﬃcient algorithm to determine a sol-

ution. Here complexity increases as some power of the number of factors.

12

While more diﬃcult (in the sense that these problems demand increasingly

greater computer resources), these problems can also be solved (assuming a

reasonable upper bound on the number of factors).

By the time a problem reaches 20 factors, the complexity for certain

P-type problems (even when a solution demonstrably exists) would

require more computer resources (processing, energy and time) than is avail-

able in a near-eternity. These tough problems are provisionally grouped in a

subset known as NP (the ‘N’suggests that these problems are ‘non-determi-

nistic’–which eﬀectively means there is no known, practical algorithm that

can shorten the pathway to an eﬃcient solution). If we are given a solution to

an NP problem, we can verify that it is correct; however, we cannot solve an

NP problem (not knowing the solution) in polynomial time.

The most diﬃcult problems have exponential solutions (such problems

belong to the complexity category known as EXP). Here the complexity of

the problem increases in a power that depends on the number of factors.

11

For a general overview, see Walter Dean, ‘Computational Complexity Theory’in Edward N Zalta (ed),

Stanford Encyclopedia of Philosophy (2016) <https://plato.stanford.edu/archives/win2016/entries/

computational-complexity> accessed 21 June 2021.

12

‘Pis the class of problems solvable by a Turing machine in polynomial time. In other words, Pis the

union, over all positive integers k, of TIME(n

k

).’Scott Aaronson, Quantum Computing since Democritus

(Cambridge University Press, 2013) 55.

LAW, INNOVATION AND TECHNOLOGY 3

As the number of factors increases from 1 to 2, the complexity is squared. As

the number moves from 2 to 3, the measure of complexity is cubed.

Computational complexity theory is a good starting point to identify pro-

blems for which quantum computers might demonstrate ‘quantum supre-

macy’. However, simply because a problem is too complex for a classical

computer, we cannot presume that quantum computers will perform any

better. Computational complexity theory –for the moment –is itself pre-

mised on the capabilities of classical computers; it will inevitably expand

to include additional categories of problems by their relative diﬃculty to

solve using quantum computers.

13

Until mathematicians identify precisely

which algorithms are tractable using quantum computers, the parameters

of ‘quantum supremacy’will remain unclear.

With respect to law, there is the added uncertainty concerning the extent

to which computational complexity theory could be used to assess legal

applications.

14

Computational law is in its infancy. Most stated legal algor-

ithms (even those deemed by lawyers to be complex) are not complex in a

computational (computer resource) sense.

15

But just as new and more

powerful algorithms emerge, computational law, too, will develop. The pro-

mulgation of law in computational form

16

and the accelerating translation of

law from human language to computer code will facilitate the advent of

complex quantum legal algorithms.

In light of these inevitable uncertainties, we outline in this essay several

categories of problems where anticipated ‘quantum supremacy’could be of

legal relevance. One of those categories concerns optimisation problems.

The law is replete with multi-factor tests, where judges or legislatures

create lists of factors that are to be considered (as a matter of process) or

weighed (as in ‘balancing tests’) before reaching a legal conclusion. These

multi-factor mandates may be translated into complex algorithms that fall

into the complexity class NP. This means that they will not be solvable

using classical computers. These legal optimisation challenges may,

however, be resolved using quantum computers (assuming appropriate

quantum algorithms can be identiﬁed for them). A second category of pro-

blems relates to law’s elaborate rules for allocating and resolving questions of

burden of proof on an operational level. Computational law will have to

properly model the operations of these rules to have the requisite power.

Burdens of proof necessarily involve the simultaneous operation of many

13

Aaronson (n 12) presents the quantum complexity class BQP, the class of problems solvable on a

quantum computer in polynomial time. ibid 136.

14

Eric Kades ﬁrst assessed the relevance of computational complexity theory to law in a 1997 article. Eric

Kades, ‘The Laws of Complexity and the Complexity of Laws: The Implications of Computational Com-

plexity Theory for Law’(1997) 49 Rutgers Law Review 403.

15

See the discussion of various juridical notions of complexity by Kades (n 14).

16

Already, US legislation must be expressed in machine-readable form. Open Government Data Act of

2019, S 760 / HR 1770.

4J. ATIK AND V. JEUTNER

factors. As such, administration of burdens of proof may be an area of com-

putational law well-suited for quantum computing. Finally, the utilisation of

quantum computing based machine learning may enhance the ability to

create legal models.

This essay addresses the prospects of these three areas for further explora-

tion. However, in order to contextualise our discussion in the essay’sﬁnal

part, we commence with an introduction of the promise of quantum com-

puting in section 2and unearth the speciﬁcally legal implications of

quantum computing for computational law in section 3.

2. The promise of quantum computing

An appreciation of the legal signiﬁcance of quantum computing presupposes

an accurate understanding of the characteristics of quantum computing

technology. Thus, this ﬁrst section of the essay brieﬂy explains the function

of quantum computers (2.1), introduces the ﬁrst identiﬁed quantum algor-

ithm (2.2) and relates quantum technology to computational complexity

theory (2.3).

2.1. The power of qubits

Already, operationalised algorithms are used in the emerging ﬁeld of legal

analytics to model, predict or instantiate the legal system. Inevitably, law

will move –at least in part –to computer code, in a transition that resembles

law’s movement from orality to text.

17

The promised range of AI-related

technologies will supplement human operators of the legal system. In this

regard, quantum computing opens up speciﬁc potentialities for compu-

tational law, enabling new (and possibly strange

18

) algorithms.

However, in order to address the question of what quantum computing

can and will bring to law, when quantum computing arrives as a deployed

technology, we must ﬁrst explore the more general question of what

quantum computing can achieve that is beyond the reach of classical

computers.

Ordinarily, a classical computer stores and processes information in

binary units, called bits. These bits can have the value of either 1 or

17

For a treatment of the signiﬁcance of law’s transition from orality to written codes, see Peter M Tiersma,

Parchment, Paper, Pixels: Law and the Technologies of Communication (University of Chicago Press,

2010). See also Rosalind Thomas, ‘Written in Stone? Liberty, Equality, Orality and the Codiﬁcation of

Law’(1995) 40 Bulletin of the Institute of Classical Studies 59.

18

Quantum phenomena, such as wave/particle duality, are often described as ‘strange’or ‘weird’. These

phenomena display characteristics or behaviors that do not correspond to ordinary human sense

experience. A ‘strange’quantum algorithm might follow a surprising pathway to reach its result.

See, for example, Daniel F Styer, The Strange World of Quantum Mechanics (Cambridge University

Press, 2000).

LAW, INNOVATION AND TECHNOLOGY 5

0. Quantum computers do not use bits, but rather qubits, to store and

process information. Qubits can be set to 1 or 0, like a classic computer.

But, importantly, they can also be set to be 1 and 0 at the same time. This

technological diﬀerence is the reason that quantum computers are vastly

more powerful than classical computers. To illustrate, one of the manufac-

turers of quantum computers recently reported that their quantum computer

performed a calculation in 1 s that would take a classic computer 10 000 years

to perform.

19

In general, it is anticipated that fully functioning quantum

computers will be 100 million times more powerful than contemporary

desktop computers and at least 3500 times more powerful than contempor-

ary super-computers. Since quantum computers operate through the

manipulation of an actual quantum system it is no surprise that they can

be usefully deployed to model other quantum systems. For example, the

folding of complex biomolecules may involve quantum mechanical pro-

cesses; and so quantum computers may facilitate the modelling of these pro-

cesses, permitting designer biomolecules.

20

It is important to appreciate at this stage that, with respect to certain pro-

blems, the potential speed of quantum computers is so superior to that of

classical computers that problems hitherto intractable by classical computers

become tractable. That does not mean, however, that quantum computers

will replace classical computers across the board. Indeed, quantum compu-

ters are not per se faster or more powerful than classical computers. Both

types of computer will co-exist, each with strengths in its own domain.

Indeed, there appear to be applications where hybrid computer systems,

involving both quantum and classical processors, may be optimum.

21

Con-

sequently, much of quantum computing is identifying which problems fall

within the class of problems that quantum computing can easily crack.

To see the possibilities for law, we have to ask whether there are legal

questions that are suited to quantum computers. There is the intriguing

possibility that law, or at least certain aspects of law, may be revealed to

have a fundamental quantised structure

22

–much as Newton’s mechanics

was revealed to be only a generalisation of more speciﬁc quantum mechanics

when taken to a large scale.

23

19

Mark Molloy, ‘Google’s New Quantum Computer Is “100 Million Times Faster than Your Pc”’ The Tele-

graph (9 December 2015) <https://www.telegraph.co.uk> accessed 21 June 2021.

20

See Carlos Outeiral and others, ‘The Prospects of Quantum Computing in Computational Molecular

Biology’(2021) 11 WIREs Computational Molecular Science.

21

See Bova, Goldfarb and Melko (n 4).

22

For a legal invocation of quantum mechanical concepts, see Valentin Jeutner, Irresolvable Norm

Conﬂicts in International Law: The Concept of a Legal Dilemma (Oxford University Press, 2017) 3. See

also Tamara Ćapeta, ‘Do Judicial Decision-Making and Quantum Mechanics Have Anything in

Common? A Contribution to Realist Theories of Adjudication at the CJEU’in Martin Belov (ed), The

Role of Courts in Contemporary Legal Orders (Eleven, 2019).

23

Intriguingly, the 1787 Constitution of the United States has, for example, been described as ‘Newtonian

in design, with its carefully counterpoised forces and counterforces, its checks and balances’. Lawrence

6J. ATIK AND V. JEUTNER

We can imagine at least one set of problems that is relevant to law where

quantum computing holds promise: complex optimisation problems. What

appear to ordinary mortals to be discrete problems, mathematicians regu-

larly see as one and the same problem.

24

Many optimisation problems are

understood to be versions of the travelling salesman’s problem (also

known as the TSP)

25

which involves optimising a pathway linking a large

set of dispersed points. Optimal designs of computer chips are solutions to

the TSP.

Imagine a travelling salesman needs to visit four cities, which are separ-

ated by various distances. Given a small number of cities (and knowing

the distances between each possible city pair), one can readily determine

the shortest pathway linking those four cities. Here one seeks to optimise

the travelling salesman’s trip by shortening it.

While the TSP can be solved quite easily for a small number of cities, it

quickly becomes intractable, as the combinations of pathways among, say

26 cities, is astronomical. This is a form of optimisation problem that

would take a classical computer ages to solve. And it is one that, at least in

theory, a quantum computer could solve in seconds.

For the moment, we do not have a general view of law as a quantum

system. Indeed, it likely is not even a classical system; ‘legal logic’only

vaguely correlates to classical logic, and there is much to law that deﬁes clas-

sical modelling. But that does not mean that a day might not come where law

is recognised to be fundamentally quantised. And if that is the case, then

quantum computers would be particularly suited to model, predict and

instantiate law. A prominent point of departure for imagining the compe-

tences of quantum computers is Shor’s Algorithm which will be explored

in the next section.

2.2. Shor’s algorithm

There are at present few identiﬁed applications where quantum computers

have a signiﬁcant advantage to classical computers. This does not mean

that there may not ultimately be many more problems identiﬁed in the

future where quantum computing will enjoy an advantage. One application

where quantum computers dominate classical computers is large number

prime factorisation.

H Tribe, ‘The Curvature of Constitutional Space: What Lawyers can Learn from Modern Physics’(1989)

103 Harvard Law Review 1, 3.

24

‘We can say that almost all of the ‘hard’problems are the same hard problem in diﬀerent guises –in

the sense that, if we had a polynomial-time algorithm for any one of them, then we’d also have poly-

nomial-time algorithms for all the rest.’Aaronson (n 14) 57.

25

See William J Cook, In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation (Prin-

ceton University Press, 2012).

LAW, INNOVATION AND TECHNOLOGY 7

Large number prime factorisation involves the identiﬁcation of the two

prime numbers that, when multiplied, generate the large number of interest.

Large number prime factorisation is well-recognised as a diﬃcult problem in

number theory. There is no known (non-quantum) algorithm available for

identifying the two prime factors that, when multiplied, give the deﬁned

large number product. In other words, there is no known direct path to iden-

tifying the factors if we have their product. Rather, one has to use slow and

laborious techniques, just slightly more eﬀective than trial and error.

Large number prime factorisation is a nice example of so-called one-way

functions

26

in mathematics. Given two primes, it is very easy to ﬁnd their

product. For example, 4507 and 7883 are both prime numbers. A schoolchild

can, through a small number of operations, compute their product:

35,528,681. A computer can reach the same product (performing the algor-

ithm for multiplication of two integers) in a fraction of a second. But things

are much more diﬃcult working in the opposite direction. It would take a lot

of trial and error to ﬁnd the prime factors of 35,528,681 if you did not already

know them –more challenging for the human as well as the computer, as

both lack a tractable algorithm that can be applied to the problem.

Utilisation of one-way functions, such as large number prime factoris-

ation, underlie much of contemporary cryptography. It is easy to ﬁnd the

result working in one direction (determining the product, given

the factors) and very diﬃcult to do so, working in the other (determining

the factors, given the product).

This well-known, very diﬃcult problem (the inherent diﬃculty of which

contributes to its usefulness) will likely yield to quantum computing. Shor’s

Algorithm

27

is the ﬁrst important quantum algorithm identiﬁed that, when

implemented on a viable quantum computer, will elegantly and eﬀectively

solve the problem of large number prime factorisation. And again, obtaining

solutions to this problem is not devoid of practical signiﬁcance: a quantum

computer may be utilised to crack the strongest cryptographic tools now

deployed.

Indeed, the only cryptographic tools that will survive the era of quantum

computers may be quantum cryptographic tools (whatever that may mean).

Cryptography is, of course, increasingly relevant to law. Cryptoeconomic

innovations –such as the blockchain, cryptocurrencies and smart contracts

–fundamentally rely on robust cryptography to substitute for trust in human

and institutional intermediaries. These promising sets of new social arrange-

ments may be struck down by the unleashed ability to crack encrypted locks.

26

See discussion of one-way functions in terms of computational complexity in Aaronson (n 14) 101–108.

Certain one-way functions yield when ‘trapdoor’information is provided. ibid.

27

Shor’s Algorithm was named after Peter W Shor. For a comprehensive introduction to the algorithm,

see Peter W Shor, ‘Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a

Quantum Computer’(1999) 41 Society for Industrial and Applied Mathematics 303.

8J. ATIK AND V. JEUTNER

The discovery of Shor’s Algorithm –and the prospect of quantum computers

that can implement it –speaks to law.

Moreover, Shor’s Algorithm and the surprising quantum solution to a

particular problem that has eluded classical computing suggests a journey

of discovery to identify currently intractable legal problems that may yield

to quantum computing.

The investigation has (at least) two essential steps:

(1) The identiﬁcation of the relevant legal problem that cannot be resolved

by classical techniques but which might be solved by quantum comput-

ing; and

(2) The speciﬁcation of a quantum algorithm that ‘solves’the legal problem.

Quantum computers, like all computers, represent capacity. To be

eﬀective they require programming. Here the input would be functional ana-

logues to Shor’s Algorithm: stepwise methods using intermediate quantum

states to output a stable solution to the legal problem.

For quantum computing to illuminate (and operationalise) law, law must

be expressed in operational terms. Thus, the ﬁrst order of business would

require creating models of legal processes. Modelling law exposes its under-

lying logic, as well as its indeterminacies. It will permit the judge and lawyer

to see law as problems seeking a solution.

Some of these problems may well be suited to solution by quantum com-

puting. But these solutions will in turn require the discovery of quantum

algorithms. We cannot know for sure the availability of a quantum solution

to a legal question until we ﬁnd it. But we must ﬁrst deﬁne the legal question.

Only if we pose a legal question in computational form can we discover

whether the utilisation of a quantum computer could bring advantages or

whether a classical computer is suﬃciently competent to resolve a given

problem. It is against that background that we now turn to a more detailed

consideration of computational complexity theory.

2.3. Computational complexity theory

Computational complexity theory has developed to capture the state-of-the-

art capacity of classical computers to solve speciﬁc categories of problems.

28

It roughly divides problems into broad classes, depending on the amount of

time (which reﬂects the number of discrete calculations employed by a par-

ticular algorithm) that a classical computer would need to output a result. As

more factors are involved in a scenario, solutions become more complex.

28

See the chapter P, NP, and Friends in Aaronson (n 14) for an accessible introduction to computational

complexity theory and the ‘zoology’of P,NP and other complexity classes.

LAW, INNOVATION AND TECHNOLOGY 9

Generally, linear problems are easiest. With respect to linear problems the

number of computer operations increases in proportion to the number of

factors involved. Polynomial functions are more complex. These are problems

where the number of computer operations increases proportionally to a par-

ticular power of the number of factors. Most complex, in this hierarchy, are

exponential functions. Here the number of operations grows by a power pro-

portional to the number of factors. As the number of factors increases, these

problems quickly exceed the capacity of classical computers to solve.

29

Computational complexity theory rates the particular class of problem by

its best-known algorithm. As such, a ‘hard’problem (one that would over-

whelm the most powerful classical computers) can be rendered ‘easy’(that

is, easily solved by a classical computer) by the discovery of a superior algor-

ithm. As such, today’s‘hard’problems may not necessarily remain so. Math-

ematicians and computer scientists are exploring new algorithmic techniques

and (occasionally or frequently, depending on your view) achieving break-

throughs. At times, these may be startling –such as reducing the best algor-

ithm from an exponential or power function to a polynomial function. At

other times, the progress in reducing complexity may be incremental.

As a science, indeed as a ﬁeld of mathematics, conventional compu-

tational complexity theory is founded upon the operational capacities of clas-

sical computers. Quantum computers bring quite diﬀerent capacities, but

these do not directly map onto the schema of classiﬁcations developed for

computational complexity theory. By engaging computational complexity

theory, we seek to identify vulnerabilities in classical computing rather

than strengths in quantum computing. But as a ﬁrst-order exercise, compu-

tational complexity theory leads us towards discoveries of ﬁelds where

quantum supremacy may lie.

3. Quantum computational law

Having outlined the essential characteristics of quantum computing above,

this section aims to sketch the potential promise of quantum computing for

computational law. In order to do so, we begin by recalling brieﬂy the charac-

ter and operation of computational law (3.1). Subsequently, we explain the

promise of marrying computational law with quantum computing (3.2).

3.1. Classic computational law

Computational law concerns the expression, application and analysis of law in

algorithmic form. It involves forming legal algorithms that proceed through

29

Aaronson (n 14) at 54 points out that the ultimate determinant of computational complexity is the

availability of a practical algorithm.

10 J. ATIK AND V. JEUTNER

logical processes (at the level of computer hardware, through passage between

‘logic gates’) to create legal conclusions. In this way, computational law

mirrors the ability of classical computers to manipulate arrays of Boolean

states. A piece of Boolean data reﬂects one of two mutually exclusive states,

such as ‘Yes’or ‘No’,or‘True’or ‘False’. Each ‘bit’(for ‘binary integer’)ina

computer records a Boolean alternative, either a ‘0’or a ‘1’. At the level of a

particular bit, computers can only see the Boolean possibilities black or

white: there are no shades of grey. Gradients can be modelled, however,

only by using more bits, digitally mixing proportions of black and white. As

computer operations are carried out, the recorded value of a particular bit

may shuttle from ‘0’to ‘1’and back. In order for law to be translated into a

computational form, at least for classical computers, it must conform to the

digital requirement of a Boolean foundation. While it is debatable whether

all law is Boolean, or binary, in nature,

30

it is certainly the case that many

legal processes can be viewed as motion from one state to another. For

example, following the logic pathway explored by Gardner,

31

one can move

from the initial state of ‘no contractual obligation’to the intermediate state

of ‘valid oﬀer’to the ultimate state of ‘contract’.

32

Traditionally, legal doctrines provide human language accounts of these

changes of states. Computational law abstracts human language renderings of

the process of contract formulation. Such abstract representations may involve

a Boolean logic tree, wherein the presence or absence of certain factors (Yes or

No responses to a Boolean query) moves the process through a logical structure

that will (or will not) reach a conclusion that constitutes a change of state.

Computational law is a new ﬁeld, but it is already evident that much law

will be expressed in machine readable code, instead of human language. In

most instances, legal reasoning is quite simple. That is, once the translation

from human language to machine readable code is completed, there is little

to computationally challenge a common computer.

However, for our purposes the question is whether and how the emer-

gence of quantum computing could actually enhance the powers of classical

computational law. We explore this question in the next section below.

3.2. Quantum-powered computational law

The emergence of quantum computers entails the potential to signiﬁcantly

enhance the powers of computational law. One of the clearest ways to

30

For a prominent theory of the binary character of legal systems, see Niklas Luhmann, Law as a Social

System (Fatima Kastner and others eds, Klaus A Ziegert tr, Oxford Socio-Legal Studies, 2004). For recent

critiques of this view, see Rostam J Neuwirth, Law in the Time of Oxymora: A Synaesthesia of Language,

Logic and Law (Routledge, 2018); Jeutner (n 24).

31

Anne von der Lieth Gardner, An Artiﬁcial Intelligence Approach to Legal Reasoning (MIT Press, 1987).

32

Ibid 124.

LAW, INNOVATION AND TECHNOLOGY 11

illustrate the potential of quantum-powered computational law relates to the

(in)famous debate concerning law’s (in)deterministic character. One of the

central claims of the Critical Legal Studies movement, and a claim frequently

discussed across many sub-ﬁelds of legal philosophy,

33

is that law is inher-

ently indeterminate.

34

Computational law presumes just the opposite: that

much of law is deterministic and can be faithfully expressed in algorithmic

form. Given certain inputs and the execution of the algorithm, a consistent

outcome is expected to result. Computational law that engages quantum

computing may reconcile these two views, permitting robust outputs while

addressing the well-recognised sources for law’s asserted indeterminacy.

Classical computing is certainly deterministic. There is no algorithm that

will produce one result one time and another result the next. Rather, classical

computation is repeatable. If law as we now know it is imperfectly determi-

nistic, then translating legal operations into computational form will result in

imposing determinism on it, which would constitute a fundamental change

to the law we have known.

But imagine the Critical Legal Studies writers have the nature of law right.

What does it mean to say that law is indeterminate? Legal writers have

noticed the simultaneous availability of contrarieties –the presence

of conﬁrmed and vital doctrines that lead to inconsistent results.

35

Each doc-

trine is fully deterministic. As such, the initial selection of which doctrine to

apply largely determines the outcome. In the realist view of law, the judge

subconsciously chose that particular doctrinal alternative (from the twin

33

In some respects, there is a remarkable similarity between the discussions concerning law’s (in)deter-

minacy and the debate among natural scientists concerning the (in)deterministic nature of the physical

world Compare, for example, Pierre Simon Laplace, A Philosophical Essay on Probabilities (Frederick

Wilson Truscott and Frederick Lincoln Emory trs, Chapman & Hall, 1902) 4; Albert Einstein, Boris

Podolsky and Nathan Rosen, ‘Can Quantum-Mechanical Description of Physical Reality Be Considered

Complete?’(1935) 47 Physical Review 777, 780; Albert Einstein, ‘Physik Und Realität’(1936) 221 Journal

of the Franklin Institute 313, 342–343; Albert Einstein, Max Born and Hedwig Born, Briefwechsel 1916–

1955 (Rowohlt 1972) 98; Niels Bohr, Atomic Theory and the Description of Nature (Cambridge University

Press, 1934) 109; Niels Bohr, ‘Discussion With Einstein on Epistemological Problems in Atomic Physics’

in J Kalckar (ed), Foundations of Quantum Physics II (1933–1958) (Elsevier, 1969); Stephen Hawking,

‘Does God Play Dice?’<https://www.hawking.org.uk/in-words/lectures/does-god-play-dice> accessed

21 January 2021.

34

See, for example, Roberto Mangabeira Unger, ‘The Critical Legal Studies Movement’(1983) 96 Harvard

Law Review 561; Charles M Yablon, ‘The Indeterminacy of the Law: Critical Legal Studies and the

Problem of Legal Explanation’(1984) 6 Cardozo Law Review 917; Lawrence B Solum, ‘On the Indeter-

minacy Crisis: Critiquing Critical Dogma’(1987) 54 University of Chicago Law Review 462.

35

Richard Nobles and David Schiﬀ, for instance, note that ‘legal statements …oscillate between contra-

dictions that cannot be accounted for through further reﬁnement’, Richard Nobles and David Schiﬀ,

‘Review of Paradoxes and Inconsistencies in the Law by Oren Perez and Gunther Teubner (eds)’

(2007) 70 Modern Law Review 505, 509. Similarly, Singer observers that ‘sometimes the best way to

express our values and our social and legal practices is by adopting what seem to be contradictory

principles, even though we cannot now, and perhaps never will, be able to reconcile them fully.’

Joseph William Singer, Entitlement: The Paradoxes of Property (Yale University Press 2000) 204–205.

For related observations concerning international law, see Martti Koskenniemi, From Apology to

Utopia: The Structure of International Legal Argument (Cambridge University Press 2005) 65.

12 J. ATIK AND V. JEUTNER

set of contrarieties) to reach an outcome desired for other, largely unprin-

cipled motives.

It may be that the observed presence of contrarieties throughout law –

with the resultant facility of an algorithm to output diﬀerent results –can

be better modelled using a quantum computer. Quantum computing incor-

porates fundamental quantum attributes, such as superposition. In quantum

physics, a superposition exists when the state of a particle (an electron, for

example) is suspended between two physical states. As a result, the particle

occupies, at least apparently, those two states at the same time. The most

famous illustration of such a superposition state involves a cat in a box

that is suspended between the state of being dead or alive.

36

In quantum

computers this phenomenon is utilised to enable signiﬁcantly faster proces-

sing of information. Strictly speaking, a superposition state is not the same as

indeterminacy. Nonetheless, in law, this phenomenon could be helpful to

conceptualise situations in which a certain kind of conduct is, as a matter

of positive law, both legal and illegal at the same time. For example, when

a contract or a treaty prohibits an act while also making it obligatory, or

when norms belonging to diﬀerent normative orders (for example, domestic

and international law) collide. Those situations occur rarely (and they are for

law an anomaly of the same magnitude as quantum physics is for classical

physics), but when they do occur references to the quantum physical super-

position phenomenon could help lawyers making sense of such otherwise

merely irresolvable norm conﬂicts.

37

Similar questions arise with respect to law’s rule / exception dynamic –a

phenomenon that has been well explored in computational legal theory.

38

Classical computing more than adequately handles the rule / exception

dynamic once the rule and any accompanying exception have been

deﬁned. But classical computing is inadequate in predicting when a new

exception will be found. The formation of a new exception constitutes a

rupture from the settled deterministic pathway. Here the phenomenon of

superposition may serve to better model this kind of legal phenomenon.

36

Erwin Schrödinger, ‘Die gegenwärtige Situation in der Quantenmechanik’(1935) 23 Naturwissenschaf-

ten 807, 812. See also Serge Haroche and Jean-Michel Raimond, Exploring the Quantum: Atoms, Cav-

ities, and Photons (Oxford University Press, 2006) 19, 27, 70–71, 82.

37

See generally, Jeutner (n 24). Ted Sichelman, in his work on the use of quantum game theory to model

the intellectual property regime, suggests that IP rights can be better described probabilistically. IP

rights lie somewhere on a continuum between the poles representing the absence of IP rights and

‘ironclad’IP rights that provide the holder with absolute protection –a metaphorical form of

quantum superposition, Ted M Sichelman, ‘Quantum Game Theory and Coordination in Intellectual

Property’(Social Science Research Network 2015) SSRN Scholarly Paper ID 1656625 <https://papers.

ssrn.com/abstract=1656625> accessed 20 June 2021.

38

For a contemporary treatment of this topic, see Luís Duarte d’Almeida, Allowing for Exceptions: A Theory

of Defences and Defeasibility in Law (Oxford University Press, 2015). See also, Glanville Williams, ‘The

Logic of “Exceptions”’ (1988) 47 Cambridge Law Journal 261; Kevin D Ashley (n 12) 58–59; Federica

Paddeu and Lorand Bartels (eds), Exceptions and Defences in International Law (Oxford University

Press, 2020).

LAW, INNOVATION AND TECHNOLOGY 13

Indeterminacy also results from the arbitrary level of abstraction that

underlies most legal reasoning. There are few legal formulas emptier than

‘decide like cases alike’. At one extreme, there are no like cases. There is an inter-

mediate level of discrimination thatmakes law work: law does assume the exist-

ence of some like cases. But diﬀerent legal systems display diﬀerent levels of

discrimination: two cases may be alike in one system, but subject to disparate

outcomes in another. The same applies within a legal system over time; two

cases may be ‘like’in one period, yet unlike in another. The level of abstraction

can move from lesser to greater magniﬁcation –and back. There is no determi-

nate legal principle that anchors the coarseness or ﬁneness of the categorisation

of cases. Here too, quantum computing may open up the possibility for better

(that is, more insightful) modelling of this complex legal phenomenon.

Finally, indeterminacy results from the uncertainty of the quantity and

quality of information that serves as inputs to a legal process (or algorithm).

Legal institutions are constitutionally starved of information. Only a minuscule

fraction of the context in which a legal question arisesis harvested and conveyed

to a tribunal. More or less information can drastically swing outcomes. Classical

law compensates for these limitations, at times, by applying presumptions for

certain contextual predicates –but the very application of these presumptions

–highly uncertain for the most part –introduces new indeterminacy at the

same time as it eliminates other indeterminacy. Computational law introduces

the prospect for a much fuller provision of information, and hence greater accu-

racy of results.Yet at the granular level, more data creates more computational

complexity. This in turn may render classical legal algorithms which function

reliably in a data-deprived setting to no longer work. Accordingly, companies

and public authoritiesare alreadyexperimenting with the use of quantumcom-

puters due to their inherent ability to consider vastly more factors in a single

computational stroke than classical computing.

For example, biomedical companies are experimenting with the use of

quantum computers to enhance radiotherapy by simulating thousands of

variables in order to devise radiation plans that avoid damaging healthy

tissue.

39

Quantum computing could also be used to enhance the AI involved

in machine learning and pattern recognition to optimise patient diagnos-

tics.

40

Moreover, car manufacturers are joining forces with quantum tech-

nology companies to improve traﬃcﬂow predictions in an attempt to

reduce congestion and optimise travel time. Public service providers use

quantum computing to model and optimise the provision of electricity

and water. In space, quantum computing is utilised to control the movement

of satellites

41

and quantum computing can also be used to optimise AI

39

Roswell Park Cancer Institute, ‘Quantum Annealing Applied to Optimization Problems in Radiation

Medicine’<https://www.dwavesys.com> accessed 16 January 2019.

40

See generally, ‘Quantum Computing Set to Revolutionise the Health Sector’(L’Atelier BNP Paribas)

<https://atelier.bnpparibas> accessed 21 June 2021.

14 J. ATIK AND V. JEUTNER

predicting the outcome of general elections.

42

Thus, the advent of quantum

computing promises to remedy contemporary legal challenges related to the

uncertainty of the quantity and quality of information by equipping compu-

tational law to deal much better with signiﬁcantly larger data sets.

Overall, quantum computing and its phenomenon of superposition allows

computational law to capture and produce more muted, more subtle legal

outputs. This possibility –together with increasing command in generated

quantum legal algorithms –may not only facilitate the making of new law

but might also shed new light on the debate concerning law’s (in)determi-

nacy. Going beyond these more foundational observations, the next

section continues to consider the potential of quantum computers for

three concrete legal processes.

4. Targets for quantum legal algorithms

Against the background of the more foundational discussion of the potential

promises of quantum computational law above, this section considers three

more concrete legal ﬁelds of application for quantum computing

concerning optimisation problems (4.1), burdens of proof (4.2) and

machine learning (4.3).

4.1. Optimisation problems

Quantum computing oﬀers much promise for the solution of multifactor

optimisation problems. Optimisation mathematics is a well explored

feature of industrial economics. A steel plant may use a variety of inputs,

such as coal, iron, and labour, each with its respective cost per unit. One

or more factors may be subject to a ﬁnite bound. Given the mix of inputs,

an optimisation analysis involves maximising the proﬁt from steel manufac-

ture, which involves ﬁnding a solution that proﬁtably maximises the output

of steel and minimises the costs of the available inputs. Such optimisation

problems are tractable when the number of factors (output of steel and

coal, iron and labour inputs) are few in number. When the number of

factors expands, determining the optimal mix involves an ever-increasing

amount of computation. Optimisation problems become signiﬁcantly

more complex when some factors are partial or total substitutes for others

or where the ratio of factors employed can vary. Moreover, additional com-

plexity can result from shifts in prices for the ﬁnished good or in the costs of

41

Gideon Bass and Booz / Allen / Hamilton, ‘Heterogeneous Quantum Computing for Satellite Optimiz-

ation’(September 2017) <https://www.dwavesys.com>.

42

Max Henderson, ‘Quantum Machine Learning for Election Modeling’<https://www.dwavesys.com>

accessed 21 June 2021.

LAW, INNOVATION AND TECHNOLOGY 15

the inputs. Quantum computing may permit far more accurate solutions to

determining the proper mix.

Law often poses optimisation challenges. Every time a judge invokes a

‘balancing test’, she is asking herself to optimise some output (quanta of

justice, perhaps) given the imposition and distribution of legal burdens on

the parties before her. Consider the four-factor test applied by an American

judge (exercising her equitable discretion) to determine whether to grant a

permanent injunction:

.plaintiﬀwould suﬀer an irreparable injury

.inadequate remedy at law

.balance of hardships

.public interest

In eBay v. MercExchange,

43

the US Supreme Court presents this four-

factor test as an ordinary judicial operation.

44

To translate this determination

into an algorithm reveals layers of computational complexity. The ﬁrst factor

–whether the plaintiﬀwould suﬀer irreparable injury –seems the easiest to

formulate. It takes a Boolean form: the judge is asked to give a ‘Yes’or ‘No’

response. But what constitutes, as a matter of law, an irreparable injury may

itself be a complex determination involving many factors.

The second factor –whether an adequate remedy is available at law –is

familiar to common law lawyers. This does not mean, however, that it

involves a simple calculation. While in many situations, the relevant prospec-

tive remedy at law is damages, the doctrine admits the possible presence of

other remedies. But even if we conﬁne the exercise to an assessment of the

adequacy of damages to remedy a harmed plaintiﬀ, there is more complexity

here than might ﬁrst meet the eye.

The third factor –the balance of hardships between plaintiﬀand defen-

dant –appears to be a fairly simple calculation, involving the quantitative

measurement of the harm to the plaintiﬀthat would result in the absence

of injunctive relief and the harm caused to the defendant should that relief

be granted. The algorithm here asks if the ﬁrst element exceeds the

second. Again, what might appear at ﬁrst to be relatively simple may turn

out to be complex. The heart of the computation is the measurement of

the two harms in play. Both involve forward looking speculation. And

each has at least some element of assessing a counterfactual situation,

depending on the duration of the behaviour sought to be enjoined. More-

over, the third factor constitutes an inequality expression. Mathematically

inequality relationships greatly increase the complexity of a relationship.

43

eBay, Inc v MercExchange, LLC, 547 US 388 (2006).

44

The eBay Court describes this four factor test as ‘well-established’and ‘traditional’.

16 J. ATIK AND V. JEUTNER

The fourth factor –consideration of the public interest –is likely beyond

translation into computational form. The various aﬀronts to the public inter-

est that might justify a refusal to grant an otherwise meritorious injunction is

limited only by the huge number of cases where such public interest con-

siderations have been evaluated.

Here are two more multifactor tests familiar to technology lawyers: the

four non-exclusive factors set out in Section 107 of the US Copyright Act

for assessing fair use

45

and the mind-boggling 15-factor test for determining

reasonable patent royalty rates enunciated in Georgia Paciﬁc.

46

Both of these

45

Section 107 of the Copyright Act of 1976 commands a judge considering an assertion of ‘fair use’to

evaluate:

(1) the purpose and character of the use, including whether such use is of a commercial nature or is

for nonproﬁt educational purposes;

(2) the nature of the copyrighted work;

(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole;

and

(4) the eﬀect of the use upon the potential market for or value of the copyrighted work.

17 USC §107.

46

See Georgia-Paciﬁc v US Plywood Corp, 318 F SUPP 1116, 1120 (SDNY 1970) for a list of factors to be

considered by a court in ﬁxing a reasonable patent royalty:

1. The royalties received by the patentee for the licensing of the patent in suit, proving or tending to

prove an established royalty.

2. The rates paid by the licensee for the use of other patents comparable to the patent in suit.

3. The nature and scope of the license, as exclusive or non-exclusive; or as restricted or non-

restricted in terms of territory or with respect to whom the manufactured product may be sold.

4. The licensor’s established policy and marketing program to maintain his patent monopoly by not

licensing others to use the invention or by granting licenses under special conditions designed to

preserve that monopoly.

5. The commercial relationship between the licensor and licensee, such as, whether they are com-

petitors in the same territory in the same line of business; or whether they are inventor and

promotor.

6. The eﬀect of selling the patented specialty in promoting sales of other products of the licensee;

the existing value of the invention to the licensor as a generator of sales of his non-patented

items; and the extent of such derivative or convoyed sales.

7. The duration of the patent and the term of the license.

8. The established proﬁtability of the product made under the patent; its commercial success; and its

current popularity.

9. The utility and advantages of the patent property over the old modes or devices, if any, that had

been used for working out similar results.

10. The nature of the patented invention; the character of the commercial embodiment of it as

owned and produced by the licensor; and the beneﬁts to those who have used the invention.

11. The extent to which the infringer has made use of the invention; and any evidence probative of

the value of that use.

12. The portion of the proﬁt or of the selling price that may be customary in the particular business or

in comparable businesses to allow for the use of the invention or analogous inventions.

13. The portion of the realisable proﬁt that should be credited to the invention as distinguished from

non-patented elements, the manufacturing process, business risks, or signiﬁcant features or

improvements added by the infringer.

14. The opinion testimony of qualiﬁed experts.

15. The amount that a licensor (such as the patentee) and a licensee (such as the infringer) would

have agreed upon (at the time the infringement began) if both had been reasonably and volun-

tarily trying to reach an agreement; that is, the amount which a prudent licensee who desired, as a

business proposition, to obtain a license to manufacture and sell a particular article embodying

LAW, INNOVATION AND TECHNOLOGY 17

formulations pack more complexity than a mere mortal judge can possibly

disentangle. It is conceivable that a quantum computer could process a

complex algorithm that pays more than judicial lip-service to these factors

and their respective interdependencies.

4.2. Burdens of proof

Much of the algorithmic structure of law takes the form of simple conditional

statements. If various factors are present (that is, if they take a Boolean

value), then a legal conclusion results or a change in legal state is eﬀected.

If only law were so simple. The conditions to be satisﬁed are frequently

quite complex themselves. Whether a valid oﬀer has been presented is a con-

dition that must be satisﬁed in order to conclude that a valid contract has

been formed.

47

However, this condition itself constitutes a legal conclusion

that algorithmically results in the satisfaction of its own set of conditions.

Legal algorithms cascade backwards, many inputs are themselves outputs

of logically prior algorithmic operations.

But there is more detail to be found within many legal algorithms. The

simple form of many algorithms involves a grammar of conditionality. In

order to determine whether a particular input is indeed present or not,

lawyers require the satisfaction of a burden of proof. The presence or

absence of a particular input may be thought of as a continuum, rather

than a mutually exclusive, bipolar distribution that signals the presence or

absence of a basis for a legal conclusion.

Some legal factors are necessary in the sense that a positive outcome

cannot result in the absence of those factors. Other factors can have positive

weight and can contribute (in a mix with other factors) to a legal outcome,

but their absence does not compel a negative result. The interplay of necess-

ary and contributory factors is more complex than would be the case were

each factor independent and necessary.

Law is replete with burdens of proof.

48

Burdens of proof are legal con-

structs that operate like step functions. Up to a certain point, the probabil-

istic presence of factors (necessary or contributory) do not throw oﬀa

change in legal state. But there is a point when one says the burden has

been satisﬁed where the legal conclusion results. Precisely where this critical

point lies is not known –but it is the conventional understanding that the

point exists and that the judge (who, at least in the Anglo-American legal

the patented invention would have been willing to pay as a royalty and yet be able to make a

reasonable proﬁt and which amount would have been acceptable by a prudent patentee who

was willing to grant a license.

47

See Gardner (n 33).

48

See the discussion of ‘proof standards’in computational models of legal arguments presented in Kevin

D Ashley (n 12) 145–146.

18 J. ATIK AND V. JEUTNER

system, assesses whether the burden of proof is satisﬁed) is able to identify it

in making a particular legal determination. A broader estimate of the pos-

ition of the critical point is obtained by collecting positive and negative

examples of satisfaction of the burden of proof within a particular category

of cases. This estimate serves both the lawyers (in their estimation of the

strengths and weaknesses of their respective cases) and the judge in

making further burden of proof determinations.

In some sense, recourse to burdens of proofs permits the judge to reach

backward into the cascade of preliminary conclusions while facing the

ultimate legal conclusion that must be rendered. Consider the ultimate

conclusion facing the judge or jury in a criminal proceeding: whether

or not the defendant has committed the oﬀence. Each deﬁned crime has

elements that function to deﬁne an algorithm leading to the imposition

of criminal responsibility. But these elements are themselves complex

and escape simple speciﬁcation (what does constitute ‘malice

aforethought’?)

As a general matter, all of the elements of a crime are essential factors, but

a judge or jury may reach the ultimate conclusion of guilt or innocence by a

mysterious recourse to the high burden of proof applied in criminal cases:

beyond a reasonable doubt.

What would a legal algorithm look like that respected both the deﬁned

elements of the crime (and the backward cascade of lower-level elements

supporting each) as well as an overall burden of proof?

Burdens of proof can also be thought of as involving mathematical

inequalities that relate all subsidiary elements. Consider the common law

crime of burglary. In its simple form, the crime has two elements: entering

a structure illegally and having the intent to commit a crime.

49

An algorithm

would depend on the presence of (1) illegal entry and (2) intent to commit a

crime. Both of these factors would have to be present in order to conclude

that a burglary took place. Each of these elements is itself a complex legal

conclusion: what constitutes an illegal entry and what constitutes intent to

commit a crime. To a large degree the two factors are interrelated: the

intent requirement refers both to the entry (in that it motivates the entry)

and to the goal of the entry. Lower level factors that suggest illegal entry

(the presence of certain tools) may also contribute to a demonstration of

intent. Here too a quantum computer may be able to fully capture the

nuanced interplay of multitudinous subsidiary factors that comprise

elements of the legal algorithm.

49

Burglary is deﬁned at common law as ‘[t]he breaking and entering the house of another in the night

time, with intent to commit a felony therein, whether the felony be actually committed or not.’

LAW, INNOVATION AND TECHNOLOGY 19

4.3. Machine learning and quantum computational law

Machine learning will be intensively utilised in many computational law

applications. The legal system presents a complex of rules, principles and per-

mitted and forbidden lines of reasoning. The law is diﬃcult enough where its

sources are conﬁned to statutes and secondary legislation; the task of model-

ling a legal system is much more diﬃcult where formal status is assigned to

decisional law, as in the Anglo-American common law system. Judicial

decisions state and re-state legal notions. Generally, but not always, these

decisions follow validated reasoning pathways. The demand to recognise

what constitutes law –for the purpose of building a model –likely exceeds

the capacity of any human lawyer. Machine learning promises the ability to

access vastly greater loads of data and to ﬁnd new and old patterns within

them. In the case of law, machine learning may be more able to ‘restate’

what the law is by including in its base a vastly greater number of decisions.

50

Machine learning has been identiﬁed as an area that may beneﬁt from

quantum computing. Machines ‘learn’by adjusting the weights given to

various input factors to create better and better predictions (that is,

outputs that can be evaluated and validated). Every exercise of a machine

is an opportunity to reﬁne its power and accuracy, through small adjust-

ments to the weights assigned to the various linkages that comprise the artiﬁ-

cial neural network. These slight adjustments are accepted when they lead to

better results; otherwise the direction of adjustment is reversed. Through a

process known as ‘gradient descent’a learning machine seeks to improve

the power of its appreciation. The artiﬁcial neural network can be rep-

resented by a system of linear equations. Together, these may generate a

function that includes local maxima and minima (peaks and valleys). A

single highest point can be found by a learning machine by divining upslop-

ing and downsloping moves. A problem results where there is more than one

peak. Here, gradient descent may bring the machine to a local peak beyond

which there is another valley. The learning machine cannot feel its way to the

even higher peak on the other side of that valley. Which is to say, it will not

reach the optimal values in every case.

Quantum computing promises to solve this problem –as it will permit the

machine to see its way beyond its current locale. Finding the optimal model

will become a more certain process if quantum computing can be deployed.

If quantum computing can bring about a general enhancement to

machine learning, computational law will certainly beneﬁt. Learning

machines eﬀectively write algorithms (though these algorithms may not be

transparent). If one wanted to say what the law is in a particular domain –

based on the entire corpus of relevant cases –machine learning will likely

50

See Introduction in Michael A Livermore, Law as Data: Computation, Text, and the Future of Legal Analy-

sis (Daniel N Rockmore ed, SFI Press, 2019).

20 J. ATIK AND V. JEUTNER

outperform the best drafting committee of the American Law Institute.

51

Quantum computing will enhance the accuracy of the resultant model by

locating the relevant optimal mix of values and weights.

5. Conclusion

The anticipated beneﬁts of quantum computing for the natural sciences and for

the global technology sector are frequently commented upon. This essay aimed

to show that quantum computing is also a legally relevant phenomenon.

52

As

established above, quantum computing technology could be utilised to greatly

enhance the abilities of the emerging ﬁeld of computational law to express,

apply and analyse law in algorithmic form. In order to advance our thesis,

we referred to certain quantum phenomena, such as superposition states,

and drew parallels between normative discussions concerning (in)determin-

ism in the natural science and in legal theory. Indeed, we share the view that

‘the metaphors and intuitions that guide physicists can enrich our comprehen-

sion of social and legal issues’

53

and that reﬂection ‘upon certain developments

in physics can help us hold on to and reﬁne some of our deeper insights into the

pervasive and profound role law plays in shaping our society’.

54

At ﬁrst sight, these attempts to invoke and refer to quantum mechanical

phenomena in a legal context might appear to be peculiar. Historically,

however, it is by no means unusual that developments in the natural sciences,

in general, and in physics, in particular, generate knock-on eﬀects across the

(perceived) natural / social science divide.

55

In the legal sphere, the emer-

gence of computational law and legal analytics is itself an example of the

fusion of ordinarily separate strands of mathematical and legal reasoning.

It is against that background that this essay has argued that legal academics

and practicing lawyers alike should reﬂect upon the legal signiﬁcance of the

unfolding ‘Second Quantum Revolution’.

In order to facilitate serious investigations into the legal utility of quantum

computing technology, we argued that two conceptual steps need to be taken:

ﬁrst, it must be identiﬁed with respect to which kind of problems quantum

computers enjoy ‘quantum supremacy’and, second, the contours of

51

Ibid.

52

For thoughts on the legal and social dimension of quantum computers, see Valentin Jeutner, ‘The

Quantum Imperative: Addressing the Legal Dimension of Quantum Computers’(2021) 1 Morals &

Machines.

53

Tribe (n 25) 2.

54

Ibid.

55

In the context of history, for example, Clark observes that the debate of physicists concerning uncer-

tainty and indeterminacy and ‘their claim that the position of the observer must be taken into account

in measuring space and time, cast doubt on the model of the omniscient observer assumed by classical

theories of knowledge’at the beginning of the 20th century. Elizabeth A Clark, History, Theory, Text:

Historians and the Linguistic Turn (Harvard University Press, 2004) 16. See also, Alexander Wendt,

Quantum Mind and Social Science (Cambridge University Press, 2015).

LAW, INNOVATION AND TECHNOLOGY 21

‘quantum supremacy’thus identiﬁed need to be related to the legal context.

With respect to the ﬁrst step, we sought to explain, with reference to compu-

tational complexity theory, how the superiority of quantum computers

as compared to classical computers relates, in principle, to problems of

NP-nature. That is, quantum computers are superior only with respect to

non-deterministic problems for which no known, practical algorithm exists

that can shorten the pathway to an eﬃcient solution. With respect to the

second step, we then demarcated the possible contours of legal ‘quantum

supremacy’by showcasing three anticipated legal ﬁelds of quantum technol-

ogy: optimisation problems, burdens of proof, and machine learning. These

three use-cases are only examples of potential applications of quantum com-

puting technology in the legal sphere. There might be many more. At the same

time, however, it should also have become apparent that the legal signiﬁcance

of quantum computers will, in any event, be limited to a speciﬁc subset of legal

questions. Indeed, we explicitly acknowledge that there are many legal issues

that (quantum) computational law cannot adequately capture.

Even with respect to the subset of legal issues that could be aﬀected by the

emergence of quantum computing technology, the exact manifestation of

quantum computational law remains, at this point in time, speculative

since the successful application of quantum computing to the ﬁeld of law

requires the concurrent evolution of two uncertain lines of development:

on the one hand, quantum computers that can work on an eﬃcient scale,

and, on the other hand, a more advanced ability to translate law from

human language to computer code. Only a convergence of these two lines

of development will ring in the era of computational law.

When these two lines of development converge is diﬃcult to predict. But

despite these uncertainties, we submit that there is merit in embarking upon

this essay’s unapologetic consideration of the legal signiﬁcance of quantum

technology in the future since the meaningful utilisation of quantum com-

puting in the legal sector at a later stage presupposes a creative imagination

of possible use-cases at the present.

Acknowledgements

The authors have explored quantum theory and law with Karl Manheim and Timo

Minssen, fellow collaborators within the Quantum Law Project at Lund. They are

grateful for comments and suggestions from Elizabeth Pollman and Ted Sichelman.

The authors enjoyed the support of colleagues at Lund and Loyola, including Xavier

Groussot, Justin Levitt, Mia Rönnmar, Michael Waterstone and Lauren Willis. Por-

tions of this essay were presented at the Second International Computational Law

Forum held at Tsinghua University School of Law, Beijing, September 21 & 22,

2019. The Quantum Law Project is funded by the Wallenberg Programme on AI,

Autonomous Systems and Software –Humanities and Society (WASP-HS).

22 J. ATIK AND V. JEUTNER

Disclosure statement

No potential conﬂict of interest was reported by the author(s).

Notes on contributors

Jeﬀery Atik is Professor of Law at Loyola Law School in Los Angeles and Guest Pro-

fessor of Civil Law at Lund University. His research focuses on law and artiﬁcial

intelligence, including computational law and national security / international

trade dimensions.

Valentin Jeutner is Associate Professor of Law at Lund University, Sweden. His

research focuses on foundational questions of (international) law. At Lund, he

serves as the PI of Sweden’s Quantum Law Project.

LAW, INNOVATION AND TECHNOLOGY 23